First off: why use this method instead of the method described by Rhenaya?

This method allows for a basically unlimited number of aquifers without having to dig ever widening rings. Using the ring method, if you have four z-layers of aquifer, your outermost ring will be huge.

This has a constant profile. The ring method gets narrower every layer.

Why use the ring method instead of this one?

If you only have 2 layers.

When you don't have the ability to make screw pumps (for instance, very limited wood or stone).

The basic principle relies on these facts:

Pumping water into a channel adjacent to an aquifer wall (where water has leaked from the aquifer) makes the water being pumped disappear (for all intents and purposes).

Water from an aquifer wall does not flow diagonally.

One pump can keep an area clear long enough to do parts of a task (for instance, building a wall).

The end result of the process is a four by two set of constructed up/down stairs surrounded by constructed walls. Once you've pierced all layers of the aquifer, you can return to digging the stairs and not building the walls.

Some basic assumptions:

You are digging a set of downward stairs, whatever profile you prefer (3x3, 4x2, 10x10, doesn't matter).

You hit the aquifer layer.

No digging has been executed on the aquifer layer yet. If any digging has resulted in a square being filled with water, congratulations, that's one side of your hole, and you shouldn't dig any more except using these instructions.

And some notes:

The images posted here are from 0.31.25, using the Phoebus graphic tileset. It's pretty simple to understand. The only things you'll need to recognize are designations for channels, stairs and screw pumps.

Your pump operators will pump even when they are hungry or thirsty. Pay attention to that so you don't lose them. If you care.

Onto the process:

Step 1: Swear at the aquifer. This isn't strictly necessary, but it makes you feel better. Some people even like them. They're wierd.

Step 2: On the z-layer ABOVE the aquifer, create a 4x2 room. The middle 2 cells of one of the long rows should contain some sort of "UP" stairs to allow dwarf access. The OTHER long row should have only floors. Down stairs will need to be covered by a constructed floor (which will be removed later). A picture for your viewing pleasure:

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Step 3: Dig 1x2 channels along the short sides of your 4x2 room.

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Step 4: Build a screw pump facing one direction on the only open floor left. You do have the components for a screw pump, don't you?

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Step 5: Pump it, baby! I mean, uh, start pumping manually (or by gears and axles, but that will require more work setting up than is worth).

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Step 6: On the z-level of the aquifer you're trying to dig through, on the side you're pumping from, opposite the pump, mine ONE SQUARE.

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Step 7: Once the miner digs that out, pause the game, and one-step by hitting period (.) until there is only 1/7 water in that mined out square.

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Step 8: Build a wall in that square.

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Step 9: Resume the game and get ready to unsuspend the building repeatedly. The mason (assuming you're using stone walls) will come over, put a stone in place, then suspend the job because the water is TOO HIGH OMGWTFBBQ! Hit "q" then "s" to unsuspend the job. The mason will stop by again, and suspend the job again. Unsuspend it. Keep doing this. If the mason touches the square in any way, shape or form, progress WILL be made. Unskilled masons can take as many as 20 unsuspends to complete the wall. FINALLY, it's done.

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Step 10: Dig 2 tiles along the long wall.

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Step 11: Pause and one-step until the mid-wall square has 1/7 water. Why this spot and not the one right next to your other wall? Because if you build the other wall, this wall will not ever be built, because the game doesn't think the mason can get to it.

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Step 12: Resume the game, and go through the Unsuspend Dance again. Eventually, the wall will be built.

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Step 13: That square you left open? Build a wall in it. You know how to do that.

Step 15: Go back up to your screw pump. Let the pump operator go get a beer. They deserve it! While they're boozing it up, deconstruct the pump, then reconstruct it facing the opposite direction (if it was pumping from the west, have it pump from the east, etc).

Step 16: Repeat step 6 through 14.

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Step 17: Deconstruct your screw pump. Now comes the fun part. Dig down stairs or a channel where the screw pump used to be. Then build a pair of up/down stairs where the screw pump used to be. You technically only need one set of stairs, but why not build 2? These will not be deconstructed, so make them out of the material you want for them.

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Step 18: Mine out your old stairs. Do NOT dig channels here, or you will ruin the whole thing.

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Step 19: If you have down stairs in these now open squares, put floors over them. They will be deconstructed later, so they can be any material.

Step 21: Get that alcoholic pump operater back to work. On the z-layer of the aquifer, mine the opposite tile again (as you did in steps 6-9).

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Step 22: Mine and wall the side tiles (like you did in steps 10-13). DO NOT UNDER ANY CIRCUMSTANCES MINE THE SQUARE UNDER THE PUMP. This is the aquifer tile that allows the construction of the OTHER side to complete.

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Step 23: Go back up a layer and reverse your pump (you were pumping from the west? Pump from the east). Build all four walls, including under the pump.

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Step 24: You can stop pumping now, no water will be coming in this side. Deconstruct the interior wall DIAGONAL from the aquifer tile that remains inside your walls (the one I told you not to dig out in step 22). Don't worry, no water will leak through that gap, aquifers only generate water orthogonally, not diagonally.

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Step 25: Mine that last square of natural wall.

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Step 26: Deconstruct your remaining interior walls. Do NOT deconstruct the walls on the outside of the 4x2 rectangle, or you will have wasted your time.

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Step 27: Build some up stairs under the up/down stairs you built on the layer above. If there is too much water on this z-layer, throw a channel along one of the short walls and it will drain down to 1/7 in all the remaining tiles.

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Step 28: If the next layer down is another aquifer, repeat steps 1 and 3-27. This time, step 1 is important.

Step 29: One thing to pay attention to: When water drains into a layer below, whether or not that layer is an aquifer, the stone will be damp. The way to tell if it's an aquifer is to observe how much water is in the dug out tile. If it's not 7/7, that layer is not an aquifer.

Step 30: Once you've gone through all your aquifer layers, you can build up-down stairs on all 8 squares of your hole for added traffic flow.

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Thanks go to QuantumMenace and Hans Lemurson on the forums for coming up with the original processes and posting them on the forum. If you have different methods than this or the ring method, let's see 'em!